A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121

Offset: 1

Author

Robert A. Russell, Sep 28 2020

Keywords

• nonn

Comments

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.

There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

Conjugacy Class Count Odd Cycle Indices

Inversion 1 x_2^4

Vertex rotation* 8 x_2^1x_6^1 Asterisk indicates that the

Edge rotation* 6 x_1^4x_2^2 operation is followed by an

Small face rotation* 3 x_4^2 inversion.

Large face rotation* 6 x_2^4

Links

• Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Crossrefs

Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).

Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).

Other polyhedra: A006003 (tetrahedron), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).

Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

Programs

Mathematica

Table[n^2(7+2n^2+3n^4)/12, {n,30}]

Formula

a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.

a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.

a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).

G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.